Difference Between Eigenvalues and Eigenvectors

Linear algebra, a significant area of mathematics that forms the foundation of many fields in physics, engineering, and even computer science, uses concepts like eigenvalues and eigenvectors. In order to solve a variety of differential equations and gain a deeper understanding of the complex aspects of a particular system, both provide a toolset about linear transformations as linear operators, as well as their properties and activity. Let's discuss the mathematical details of eigenvalues and eigenvectors and highlight their applications and differences in this article.

What are Eigenvalues?

The numerical values associated with the linear transform's eigenvectors are known as eigenvalues. The German word Eigen, which means characteristic, is the source of the English word Eigen, which is used in mathematics. Thus, these are eigenvalues, which indicate how much eigenvectors are extended in the eigenvector's direction. Except in cases where the eigenvalue is negative, it does not assume that the vector's orientation has changed. The orientation is simply reversed when the Eigenvalue is negative. The equation for eigenvalue is given

• Av = ?v
• Where,
  
  A is the matrix,
  v is associated eigenvector, and
  ? is scalar eigenvalue.

Applications of Eigenvalues

• Stability Analysis in Dynamical Systems: The stability of equilibrium points in various dynamic systems is ascertained using eigenvalues.
• Principal Component Analysis (PCA) is a technique used for variable reduction that falls within the exploratory factor analysis category.
• Quantum Mechanics: Eigenvalues are phases of a physical system that correlate to a specific value of an operator, such as the energy of an electron in an atom, according to quantum mechanics.
• Markov Chains: For defining the activity of the dynamical system, the eigenvalues of a Transition Probability matrix are used to express the measure of the distances between states in Markov chains.

What are Eigenvectors?

An eigenvector can be defined as a matrix A as "v" that satisfies the following condition. Eigenvectors for square matrices can be expressed as non-zero vector values such that the scaler multiple of the vector is the resultant matrix when the vectors are multiplied by the square matrices. Av = ?v. The eigenvalue of this square matrix is the scaler multiple ?. Finding the square matrix's eigenvalues is usually always necessary before determining the matrix's eigenvectors.

Eigenvector Equation

The formula for determining the eigenvector of any square matrix is known as the Eigenvector equation. The equation for the eigenvector is,

• Av = ?v
• Where,
  A is the given square matrix,
  v is the eigenvector of matrix A, and
  ? is any scaler multiple.

Applications of Eigenvectors

• Molecular Orbital Theory: In the molecular orbital theory of chemistry, the molecular orbitals are given by the eigenvectors of the Hamiltonian matrix.
• Mode Shapes and Mechanical Vibrations: Eigenvectors are employed in structural dynamics to represent the mode shapes of vibrating systems. These mode forms describe the structure's distortion or vibration pattern at each natural frequency, which is important information for designing and analysing structures that are affected by dynamic loads.
• Image Compression: Singular Value Decomposition (SVD) compression methods use the image matrix's eigenvectors to reconstruct the image using less information.
• Google PageRank Algorithm: The eigenvector of the adjacency matrix of the Web page graph defines the PageRank algorithm, which Google use to sort Web sites.

Difference between Eigenvalues and Eigenvectors

The following table highlights the major difference between Eigenvalues and Eigenvectors -

Conclusion

Fundamental concepts in linear algebra which help in understanding linear transformations and their effects on vector spaces are eigenvalues and eigenvectors. Consequently, eigenvectors, which are vectors, indicate the direction of this scaling, and eigenvalues, which are scalars, quantify the extent of this scaling. Stability analysis, quantum physics, vibration analysis, and data reduction methods like principal component analysis are among its applications. We learn more about their behaviour in the systems and matrices we encounter due to their long-standing descriptions.